Newspace parameters
Level: | \( N \) | \(=\) | \( 1925 = 5^{2} \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1925.w (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.960700149319\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 385) |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.2695.1 |
Artin image: | $S_3\times C_{12}$ |
Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).
\(n\) | \(276\) | \(1002\) | \(1751\) |
\(\chi(n)\) | \(\zeta_{12}^{4}\) | \(1\) | \(-1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1451.1 |
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−0.866025 | − | 0.500000i | 0 | 0 | 0 | 0 | − | 1.00000i | 1.00000i | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||
1451.2 | 0.866025 | + | 0.500000i | 0 | 0 | 0 | 0 | 1.00000i | − | 1.00000i | 0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||
1726.1 | −0.866025 | + | 0.500000i | 0 | 0 | 0 | 0 | 1.00000i | − | 1.00000i | 0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||
1726.2 | 0.866025 | − | 0.500000i | 0 | 0 | 0 | 0 | − | 1.00000i | 1.00000i | 0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
55.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-55}) \) |
5.b | even | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
11.b | odd | 2 | 1 | inner |
35.j | even | 6 | 1 | inner |
77.h | odd | 6 | 1 | inner |
385.q | odd | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(1925, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{2} + 1 \)
$3$
\( T^{4} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 1)^{2} \)
$11$
\( (T^{2} + T + 1)^{2} \)
$13$
\( (T^{2} + 1)^{2} \)
$17$
\( T^{4} - 4T^{2} + 16 \)
$19$
\( T^{4} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( (T^{2} - T + 1)^{2} \)
$37$
\( T^{4} \)
$41$
\( T^{4} \)
$43$
\( (T^{2} + 1)^{2} \)
$47$
\( T^{4} \)
$53$
\( T^{4} \)
$59$
\( (T^{2} + T + 1)^{2} \)
$61$
\( T^{4} \)
$67$
\( T^{4} \)
$71$
\( (T + 1)^{4} \)
$73$
\( T^{4} - T^{2} + 1 \)
$79$
\( T^{4} \)
$83$
\( (T^{2} + 1)^{2} \)
$89$
\( (T^{2} + T + 1)^{2} \)
$97$
\( T^{4} \)
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